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Diophantine And Almost Fermat (Posted on 2007-02-21) Difficulty: 3 of 5
Consider three positive integers x< y< z in arithmetic sequence.

Determine analytically all possible solutions of each of the following equations:

(I) x3 + z3 = y3 + 10yz

(II) x3 + y3 = z3 - 2, with y< 116.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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possible solutions | Comment 2 of 4 |
I didn't get the meaning of "arithmetic sequence" at first.  Thanks for the additional input.

I was unable to find a solution that fit both equations.  For the first equation, (1,4,7) and (3,6,9) fit.  For the second equation, (5,6,7) works.

For a solution that fits both equations, I had first tried by combining the equations to get [x=5yz-1].  From this, I determined that x must end in either 4 or 9 since 5yz ends in either 0 or 5.  Subtracting the two equations yields [z-y=5yz+1].  Using a for the arithmatic sequence step leads to [3ax+9ax+7a=5x+3ax+2a].  However, I could not find an integer solution that satisfied both equations.
  Posted by hoodat on 2007-02-26 18:20:24
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